What is a "classically chaotic quantum system"? In the context of quantum mirages one can find increased probability density around paths of unstable classical periodic orbits, called quantum scarring. In this wikipedia article they use the term "classically chaotic quantum system". What is such a system precisely?
 A: There are two types of quantum chaos, one is quantum single-body chaos while the other is quantum many-body chaos. The difference is what's the number of degrees of freedom in the system.
For example, eigenmode for some billiard systems like deformed Whispering-gallery mode exhibit chaotic behavior, but the number of DOF is only 2. On the other hand, a chaotic quantum spin chain involves $N$ degrees of freedom which manifest the many-body nature.
The so-called classically chaotic quantum system usually refers to a quantum system whose Hamiltonian (or equation of motion) has a very obvious classical counterpart and exhibits chaos. Such as the deformed WGM I mentioned earlier. The scar modes of WGM cavities are studied thoroughly during the past decades.
The quantum spin chain (for spin-1/2), on the other hand, does not have a very clear classical counterpart because the quantum fluctuation is dominating the dynamics. For these kinds of systems, however, some symmetry or emerged trajectories have mean-field (classical) dynamics. These special trajectories are called quantum many-body scars.
